position of the camera be lowered by a small amount. This amount was very small and
considered negligible in the further derivation of the vision geometry. The camera was
assumed to be positioned so that the center of its lens was at the origin of frame 3. Therefore,
the coordinate frame of the camera was the same as frame 3.
The relationship between the position of the fruit in the camera frame, Pc, and the
position of its image on the imaging array, pi, was important in formulating the effects of
changes in the position of the manipulator's joints on the position of the image. To vision-servo
the robot, it was necessary to develop closed-form solutions that related changes in 81 and 82 to
changes in xi and yi. These relationships were referred to as vision gains and expressed as
K -dyi
=d8 and (5-7)
dxi
Kvx=
d22. (5-8)
Because the position of the fruit was assumed to remain the same regardless of the
configuration of the robot, it became necessary to define the position of the fruit with respect to
the stationary base frame (see Figure 5.4). Using the T3 matrix to specify the position of the
fruit in the coordinates of the base frame, po, was established as
Po= T3Pc. (5-9)
Utilizing this relationship, the position of the fruit in the camera coordinate frame was
determined to be
-1
Pc= T3 Po. (5-10)
Substituting pc into equation 5-5, the position of the fruit's projection in the image array, pi, was
found
-1
Pi= TT3 PO.
(5-11)